Theorem cantnfres 9645

Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfrescl.d	⊢ (𝜑 → 𝐷 ∈ On)
cantnfrescl.b	⊢ (𝜑 → 𝐵 ⊆ 𝐷)
cantnfrescl.x	⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
cantnfrescl.a	⊢ (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t	⊢ 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m	⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)

Assertion

Ref	Expression
cantnfres	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛

Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres

Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfrescl.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ On)
2		cantnfrescl.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
3		cantnfrescl.x	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
4	1, 2, 3	extmptsuppeq 8183	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
5		oieq2 9474	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅) → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
6	4, 5	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
7	6	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝜑 → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
8	7	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
9	8	oveq2d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
10		suppssdm 8172	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ dom (𝑛 ∈ 𝐵 ↦ 𝑋)
11		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
12	11	dmmptss 6243	. . . . . . . . . . . . . 14 ⊢ dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵)
14	10, 13	sstrid 3956	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
15	14	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
16		eqid 2769	. . . . . . . . . . . . . 14 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
17	16	oif 9491	. . . . . . . . . . . . 13 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)):dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))⟶((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)
18	17	ffvelcdmi 7079	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
19	18	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
20	15, 19	sseldd 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ 𝐵)
21	20	fvresd 6902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
22	2	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵 ⊆ 𝐷)
23	22	resmptd 6043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵) = (𝑛 ∈ 𝐵 ↦ 𝑋))
24	23	fveq1d 6884	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
25	8	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
26	21, 24, 25	3eqtr3d 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
27	9, 26	oveq12d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) = ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))))
28	27	oveq1d 7426	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧))
29	28	mpoeq3dva 7488	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
30	6	dmeqd 5896	. . . . . 6 ⊢ (𝜑 → dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
31		eqid 2769	. . . . . 6 ⊢ On = On
32		mpoeq12 7484	. . . . . 6 ⊢ ((dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
33	30, 31, 32	sylancl 597	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
34	29, 33	eqtrd 2804	. . . 4 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35		eqid 2769	. . . 4 ⊢ ∅ = ∅
36		seqomeq12 8440	. . . 4 ⊢ (((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
37	34, 35, 36	sylancl 597	. . 3 ⊢ (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
38	37, 30	fveq12d 6889	. 2 ⊢ (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
39		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
40		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
41		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
42		cantnfres.m	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)
43		eqid 2769	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
44	39, 40, 41, 16, 42, 43	cantnfval2 9637	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))))
45		cantnfrescl.t	. . 3 ⊢ 𝑇 = dom (𝐴 CNF 𝐷)
46		eqid 2769	. . 3 ⊢ OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
47		cantnfrescl.a	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝐴)
48	39, 40, 41, 1, 2, 3, 47, 45	cantnfrescl 9644	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
49	42, 48	mpbid 235	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)
50		eqid 2769	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
51	45, 40, 1, 46, 49, 50	cantnfval2 9637	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
52	38, 44, 51	3eqtr4d 2814	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294   ↦ cmpt 5196   E cep 5561  dom cdm 5662   ↾ cres 5664  Oncon0 6361  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   supp csupp 8155  seq_ωcseqom 8433   +_o coa 8449   ·_o comu 8450   ↑_o coe 8451  OrdIsocoi 9470   CNF ccnf 9629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-seqom 8434  df-1o 8452  df-oadd 8456  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-oi 9471  df-cnf 9630

This theorem is referenced by:  cantnf2  43943